Explicit formula for N_n(F) over finite fields

Develop an explicit formula for the number N_n(F) of isomorphism classes (under T ~ S induced by A(T) ≅ A(S)) when F is a finite field, as a function of n and the size of F.

Background

After introducing invariants (the Wall and the Measure sequence) and proving the lower bound N_n ≥ C_{n−1}, the authors note that exact enumeration remains difficult, particularly for larger n because related systems of equations become complicated.

They ask specifically for a closed-form expression for N_n(F) when F is finite, which would complement their lower bounds and the finiteness results in this case.

References

This facts motivated us to open the following new questions, that we expect to answer in future works:

Q3: If is a finite field. Is there any formula for the number N_n( )?

— Invariants for isomorphism classes in the category $\bcalNT$  (2508.00084 - Maturana, 31 Jul 2025) in Introduction (Section 1)

Let us now mention a few open questions following our investigations here. Firstly, the problem of classifying and enumerating finite Alexander quandles becomes open once again.

— Multivariate Quandles as Groupoid Invariants  (2609.30262 - Arsiwalla et al., 24 Sep 2026) in Section 8, Conclusions and Discussion

\cref{rem:when psd 5 polyn} and \cref{prop: psd 5 in M2(Fq)} motivate us to conjecture the following. For a prime p and positive integer n, there exists a positive integer t and rational polynomials f_0(x),\dots,f_{t-1}(x)\in \mathbb{Q}[x] of degree \binom{n+1}{2}, such that the number of matrices in $\PSD_5(n)$ over $\mathbb{F}_{pm}$ is $f_i(pk)$ when $k\equiv i\pmod{t}$.

— Positive definite, positive semidefinite and totally positive matrices over finite fields  (2608.17702 - Ayyer et al., 18 Aug 2026) in Conjecture following Remark 2.??, Section 2, subsection “Type 5” of Section 2